1. Field of the Invention
The present invention relates to an apparatus and a method for estimating optical wavefront parameters, such as wavefront shape and aberration.
2. Description of the Related Art
Wavefront measurements are typically used to test the quality of optical surfaces and evaluate optical performance of optical elements. Wavefront measurements are also used for alignment of high-accuracy optical systems. A wavefront is the locus (a line, or in a wave propagating in 3 dimensions, a surface) of points on which all light rays have the same phase. The simplest form of a wavefront is that of a plane wave, where rays of light are parallel to each other and strike a sensor with a planar wavefront. Propagation of a wavefront through optical elements, such as lenses and mirrors, generally changes the shape of the wavefront due to lens thickness, imperfections in surface morphology, variations in refractive index, and other factors. Undesired changes in the shape of the wavefront are known as aberrations. Thus, knowledge of the wavefront profile and correction of aberrations thereof are very important when designing optical elements, and evaluating the performance of newly designed optical systems. For example, before assembling a complete imaging system, it is necessary to verify performance of each optical unit (unit lens) included in such a system. Since each unit lens or single lens itself may have certain aberrations, it is necessary to control the quality of imaging lenses with high precision. One application for measuring large wavefront aberrations is in the testing of the center of curvature of aspheric optical elements, such as lenses or mirrors.
A conventional method of measuring the wavefront quality of a light beam employs interferometric wavefront sensors in which spatial filtering of a small portion of the light source beam is used to produce a spherical reference wave that is subsequently combined with the original wavefront to produce an interferogram. As it is well understood in the art, interference fringes in the interferogram can be analyzed to evaluate the quality of the light beam. However, dividing the original beam and recombining it with the reference beam tends to introduce system aberrations, for example due to optical-path errors and improper alignment of optical components. An alternative conventional method of wavefront measurement uses non-interferometric wavefront sensors (NIWFS), such as a Shack-Hartman wavefront sensor, which do not require dividing and recombining the original beam.
NIWFS, such as Shack-Hartmann wavefront sensors (SHWFS), have a greater dynamic range than interferometric sensors. One basic and commonly used configuration of a SHWFS sensor consists of a lenslet array and an optical detector (typically a CCD camera) located at the back focal plane of the lenslet array. The Shack-Hartmann wavefront sensor divides the wavefront of an incident beam being measured into a plurality of beamlets by using a two-dimensional (2D) lenslet array. Each lenslet in the lenslet array generates a separate and independent focus (spot) on the surface of the optical detector. The centroid position of each spot is displaced by a distance and direction indicative of the wavefront aberrations between a reference and distorted beam. Therefore, wavefront measurement by a SHWFS is based on an estimation of the local slopes of the aberrated wavefront relative to a reference (plane) wavefront. Generally, the wavefront estimation procedure may be categorized as either zonal or modal, depending on whether the phase is presented as a set of local slopes of the wavefronts or as a set of coefficients of some modal functions determined across the whole aperture. In the latter, displacements of focal spots can be represented in terms of Zernike polynomials.
There are several advantages to using SHWFS over interferometric counterparts. SHWFS have greater dynamic range than interferometric sensors. The incident radiation does not have to be coherent. Since the SHWFS can acquire all of the wavefront information from a single image, exposure times can be short, which reduces sensitivity to vibration. More importantly, both irradiance and phase distributions can be obtained with a SHWFS.
FIG. 1 illustrates an example of the configuration of a wavefront measurement apparatus 100 using a SHWFS. As illustrated in FIG. 1, the wavefront measurement apparatus 100 may typically include, a light source 102, a neutral density (ND) filter 104, a beam expander 106, a projection lens 108, test optics (test element or sample) 110, a lens group 112, a lenslet array (micro lens array) 114, an optical detector 116 (CCD sensor), and a data analyzer 118. Arranged in this manner, the wavefront measurement apparatus 100 is configured to characterize the effects that the sample (test optics 110) exerts on a wavefront of light that travels through it.
The optical configuration of a SHWFS is illustrated with more detail in FIG. 2. In FIG. 2, the optical detector 116 includes a sensor array 210, whereupon light passed through lenslet array 114 is incident. Locations of focal spots 250, 251, 252, 253 and 254 on the sensor array 210 are dependent on a local tilt of an incoming wavefront 150. The local tilt of the wavefront 150 is caused by aberrations in the test optics 110 in FIG. 1. The local tilt can be calculated by variation of focal spot location. The wavefront can be reconstructed by using the local tilt information obtained from lenslets of the lenslet array 114.
When the amount of wavefront deviation is within the dynamic range of the SHWFS, positions of each focal spot on the sensor array 210 can be detected separately and assigned to the correct lenslet; and a wavefront profile can be easily determined. However, if the wavefront deviation exceeds the dynamic range of the SHWFS, as illustrated in FIG. 3, the sensor array SHWFS cannot analyze the wavefront anymore. Specifically, in FIG. 3, the lenslet array 114 forms focal spots 350, 351, 352, 353 and 354. However, due to a highly aberrated wavefront 160, the focal spot 350 extends beyond the surface range of the sensor array 210; and focal spots 353 and 354 have crossed each other (spot crossover). That is, the beams forming the focal spots 353 and 354 overlap each other and are focused at crossover locations on the sensor array 210. In the situation illustrated in FIG. 3, since focal spot 350 is beyond the range of sensor array 210, focal spot 350 cannot be detected; and because of spot crossover, focal spots 353 and 354 cannot be unambiguously assigned.
FIG. 4 illustrates an example of output data from the sensor array 210 upon detecting an aberrated wavefront. In FIG. 4, a region of interest S, showing focal spots at the outer region thereof, has been enlarged to better illustrate the effects caused by focal spots crossing over each other or extending beyond the effective area of sensor array 210.
Techniques for extending the dynamic range of the SHWFS or for analytically compensating for wavefront aberrations have been previously proposed and continue to be developed. Some of the more popular techniques are summarized below.
(1) Null Lens
A null lens includes a set of optics specifically designed to compensate or nullify an expected wavefront aberration. Since the null lens technique can completely compensate the wavefront aberration of test optics (provided that there are no manufacturing errors), it can effectively cancel wavefront deviation on the lenslet array. However, with this technique, it is necessary to fabricate a highly accurate null lens for an accurate measurement. Therefore, the fabrication cost of a null lens can become prohibitively expensive. Furthermore, such a null lens is designed for specific test optics with an expected wavefront aberration, thus this technique may be not applicable to other wavefronts formed by test optics of different shapes or characteristics. An example of the null lens technique is described in U.S. Pat. No. 5,233,174 to Zmek, which is incorporated herein by reference.
(2) Estimation Techniques
Instead of the null lens technique, wavefront estimation techniques have been proposed for measuring aberrated wavefronts, and a corrective algorithm is typically used to correct or compensate for aberrations. One example of an estimation technique is disclosed in Ref. 1: Michael C. Roggemann, Timothy J. Schulz, Chee W. Ngai, and Jason T. Kraft, “Joint processing of Hartmann sensor and conventional image measurements to estimate large aberrations: theory and experimental results,” Appl. Opt. 38, pp. 2249-2255 (1999).
Another wavefront estimation technique uses Maximum Likelihood Estimation (MLE) for wavefront reconstruction. An example of the MLE technique is disclosed in Ref. 2: Harrison H. Barrett, Christopher Dainty, and David Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A. 24, 391-414 (2007).
(3) Stitching
U.S. patent application publication No. 2009/0284753 describes a technique in which a series of wavefront measurements are “stitched” together using mathematical methods. For each measurement, a different focus, wavefront tilt or reference aberration is used in conjunction with a dynamic-range-limiting aperture. Purportedly, this technique can effectively extend the dynamic range of the sensor. However, those of ordinary skill in the art should readily appreciate that performing a series of wavefront measurements and stitching those measurements together using mathematical methods is a computing-intensive process.
In view of the foregoing state of the art, the inventors herein have developed a novel technique for estimating wavefront parameters that can, among other advantages, shorten the time required for the measurement and analysis of such parameters while correctively canceling wavefront errors.